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<h1 id="1-二叉树定义"><a href="#1-二叉树定义" class="headerlink" title="1.二叉树定义"></a><font size=3>1.二叉树定义</font></h1><p>什么是二叉树，二叉树也是一种树形结构，它是一种特殊的树，它有以下特点：</p>
<p>（1）二叉树是有序树，严格区分左孩子和右孩子，即使只有一个子节点也要区分左右；</p>
<p>（2）树中包含的各个结点的度不能超过 2 ，只能是 0 、 1 或者 2 。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502102051552.png" alt="image-20220502102051552" style="zoom:33%;" />

<h1 id="2-二叉树形态"><a href="#2-二叉树形态" class="headerlink" title="2.二叉树形态"></a><font size=3>2.二叉树形态</font></h1><p>二叉树的形态一共可以有五种：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502102603884.png" alt="image-20220502102603884" style="zoom:33%;" />



<h1 id="4-二叉树性质"><a href="#4-二叉树性质" class="headerlink" title="4.二叉树性质"></a><font size=3>4.二叉树性质</font></h1><p>（1）二叉树的第 i (i ≥ 1) 层上至多有$2^{i-1}$个结点。</p>
<blockquote>
<p>【证明】数学归纳法：</p>
<ol>
<li>i &#x3D; 1时，只有一个根节点，显然$2^0&#x3D;1$，上述结论成立；</li>
<li>假设对所有的 j ，1 ≤ i ＜j 时上述结论成立，即在第 j 层上至多有 $2^{j-1}$个结点；</li>
<li>所以在第 i - 1 层上至多有$2^{i-2}$个结点，由于二叉树的每个结点的度数至多为 2 ，所以在第 i 层上结点最多为 i - 1层上结点数的两倍，即第 i 层上结点数至多为 $2^{i-1}$。</li>
</ol>
</blockquote>
<p>（2）深度为 k (k ≥ 1) 的二叉树至多有$2^{k}-1$个结点。</p>
<blockquote>
<p>【证明】</p>
<p>由性质（1）可知，二叉树第 i 层上至多有 $2^{i-1}$个结点，一共有 k 层，所以总的结点数 n 为：</p>
<p>$n &#x3D; 2^0 + 2^1 + …… +2^{k-1}$</p>
<p>等比数列求和公式为：$S_n &#x3D; \frac{a_1-a_nq)}{1-q}$</p>
<p>所以 $ n &#x3D; \frac{1-2^{k-1}*2}{1-2} &#x3D; 2^k-1$</p>
</blockquote>
<p>（3）对任意的二叉树 T ，若其叶子结点数为$n_0$，度数为 2 的结点数为$n_2$，则$n_0 &#x3D; n_2 + 1$。</p>
<blockquote>
<p>【证明】</p>
<p>设$n_1$为二叉树 T 中度数为 1 的结点数，n 为二叉树的总结点数，则 $n &#x3D; n_0 + n_1 + n_2$ </p>
<p>对于每一个结点来说都是由结点的父结点分支表示的，假设二叉树中树的分支数为 B ，那么总结点数 $n &#x3D; B + 1$</p>
<p>而二叉树的分支数 $B &#x3D; n_1 + 2 * n_2$，所以有$n &#x3D; n_1 + 2*n_2 + 1$</p>
<p>所以$n &#x3D; n_1 + 2*n_2 + 1 &#x3D; n_0 + n_1 + n_2 $</p>
<p>两边消去$n_1$可以得到$n_0 &#x3D; n_2 + 1$。</p>
</blockquote>
<h1 id="3-几种二叉树"><a href="#3-几种二叉树" class="headerlink" title="3.几种二叉树"></a><font size=3>3.几种二叉树</font></h1><h2 id="3-1满二叉树"><a href="#3-1满二叉树" class="headerlink" title="3.1满二叉树"></a><font size=3>3.1满二叉树</font></h2><p>如果二叉树中除了叶子结点，每个结点的度都为  2 ，则此二叉树称为满二叉树。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502111938323.png" alt="image-20220502111938323" style="zoom:33%;" />

<p>满二叉树除上述三条一般二叉树的性质之外，还有如下性质：</p>
<p>（1）满二叉树的第 i (i ≥ 1) 层上有$2^{i-1}$个结点。</p>
<p>（2）深度为 k (k ≥ 1) 的满二叉树有$2^{k}-1$个结点，叶子数为$2^{k-1}$。</p>
<p>（3）满二叉树中不存在度为  1  的节点，每一个分支点中都两棵深度相同的子树，且叶子节点都在最底层。</p>
<p>（4）具有  n  个节点的满二叉树的深度为$\log_2(n+1)$。</p>
<ul>
<li>完全二叉树</li>
</ul>
<p>只有最下面一层有度数小于 2 的节点，且最下面一层的叶节点集中在最左边的若干位置上的二叉树。也就是说完全二叉树去除最下边一层的所有节点后是一个满二叉树。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502120912000.png" alt="image-20220502120912000" style="zoom:33%;" />

<p>完全二叉树除上述三条一般二叉树的性质之外，还有如下性质：</p>
<p>对于任意一个含有 n 个结点的完全二叉树来说，如果将含有的结点按照层次从左到右依次标号，如上图所示，根节点为 1 号节点，则对任意一个结点 i 有：</p>
<p>（1）当 i ＞ 1 时，结点  i 有父结点，且编号为 i &#x2F; 2 。</p>
<p>（2）当 2i ≤ n 时，有左孩子，且左孩子编号为 2i ；当 2i &gt; n 时，结点肯定没有左孩子，节点本身为叶子结点。</p>
<p>（3）当 2i + 1 ≤ n 时，有右孩子，且右孩子编号为 2i + 1 ；当 2i + 1 &gt; n 时，结点肯定没有右孩子。</p>
<p>（4）当  i  为奇数且不为 1 时，结点 i 有左兄弟，左兄弟编号为 i - 1 ，否则 i 没有左兄弟。</p>
<p>（5）当 i 为偶数且小于 n 时，结点 i 有右兄弟，右兄弟编号为 i + 1 ，否则 i 没有右兄弟。</p>
<h1 id="4-存储结构"><a href="#4-存储结构" class="headerlink" title="4.存储结构"></a><font size=3>4.存储结构</font></h1><p>我们前边学的有顺序存储结构和链式存储结构，他们都可以作为二叉树的存储结构。</p>
<h2 id="4-1顺序存储结构"><a href="#4-1顺序存储结构" class="headerlink" title="4.1顺序存储结构"></a><font size=3>4.1顺序存储结构</font></h2><p>使用顺序表（数组）存储二叉树，这叫做二叉树的顺序存储，但是要注意的就是，<strong>顺序存储只适用于完全二叉树</strong>。如果我们想顺序存储普通二叉树，需要提前将普通二叉树转化为完全二叉树。</p>
<p>那普通二叉树如何转化为完全二叉树呢？如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502141503664.png" alt="image-20220502141503664" style="zoom:33%;" />

<p>这样一来，二叉树就都可以满足顺序存储的条件了，那怎么存储呢？如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502142800579.png" alt="image-20220502142800579" style="zoom:33%;" />

<p>通过顺序表存储的二叉树，可以很轻松还原，但是顺序存储结构是特别浪费空间的，那些为了转化为完全二叉树而补充出来的结点并没有什么实际用处，反而还浪费了很大的空间，比如上图中，原本结点只有三个，但是存储的时候却至少需要多出来三个结点的空间。</p>
<h2 id="4-2链式存储结构"><a href="#4-2链式存储结构" class="headerlink" title="4.2链式存储结构"></a><font size=3>4.2链式存储结构</font></h2><p>我们还可以通过链表来存放二叉树，存储方式如下：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/image-20220502151008029.png" alt="image-20220502151008029" style="zoom:33%;" />

<p>存放结点的链式节点结构有三部分组成：</p>
<ul>
<li>Lchild ：指针域，指向存放左孩结点的链表节点的指针。</li>
<li>data ：数据域，结点存放的数据。</li>
<li>Rchild ：指针域，指向存放右孩结点的链表节点的指针。</li>
</ul>
<h1 id="5-二叉树遍历"><a href="#5-二叉树遍历" class="headerlink" title="5.二叉树遍历"></a><font size=3>5.二叉树遍历</font></h1><h2 id="5-1遍历算法的由来"><a href="#5-1遍历算法的由来" class="headerlink" title="5.1遍历算法的由来"></a><font size=3>5.1遍历算法的由来</font></h2><p>对于一个二叉树，在我们不知道有哪些遍历算法的情况下，很容易我们会想到一种，就是按层次，从上到下，每一层次从左到右来遍历，这种遍历的方式被称之为层次遍历。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/1.gif" alt="1" style="zoom:50%;" />

<p>那还有其他的遍历方法吗？还有一种就是我们遵循从上到下，从左到右的遍历方式，如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/2.gif" alt="2" style="zoom:50%;" />

<p>这样依然可以访问到所有的结点，他们的区别是什么呢？层次遍历中，每次只经过结点一次，而下边这种遍历，对于结点 B 来说，遍历的时候经过了它三次，即便是叶子结点，看上去是两次，但是也可以看做是三次。我们可以在经过这个结点不同的时机进行访问，如此便有三种访问方式了：</p>
<ul>
<li>遇到一个结点，先访问，再遍历左右子树，称之为<strong>先序遍历</strong>。</li>
<li>遇到一个结点，第一次先不访问，先访问它的左子树，再访问这个结点，最后访问右子树，称之为<strong>中序遍历</strong>。</li>
<li>遇到一个结点，第一次不访问，先访问左子树，第二次还是不访问，然后访问右子树，最后访问这个结点，称之为<strong>后序遍历</strong>。</li>
</ul>
<h2 id="5-2层次遍历"><a href="#5-2层次遍历" class="headerlink" title="5.2层次遍历"></a><font size=3>5.2层次遍历</font></h2><p>层次遍历过程如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/3.gif" alt="3" style="zoom:50%;" />

<h2 id="5-3先序遍历"><a href="#5-3先序遍历" class="headerlink" title="5.3先序遍历"></a><font size=3>5.3先序遍历</font></h2><p>先序遍历过程如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/4.gif" alt="4" style="zoom:50%;" />

<h2 id="5-4中序遍历"><a href="#5-4中序遍历" class="headerlink" title="5.4中序遍历"></a><font size=3>5.4中序遍历</font></h2><p>中序遍历过程如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/5.gif" alt="5" style="zoom:50%;" />

<h2 id="5-5后序遍历"><a href="#5-5后序遍历" class="headerlink" title="5.5后序遍历"></a><font size=3>5.5后序遍历</font></h2><p>后序遍历过程如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-11-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E7%A1%80/img/6.gif" alt="6" style="zoom:50%;" />


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